Oversimplified: Signals and Systems (4) – How to represent a signal?

A system can be thought of as a function that maps the input to an output. In this class, most of the systems that we are interested in are linear, therefore the functions are linear as well so superposition applies.

We haven’t defined what the inputs are. In fact it’s arbitrary, but in ‘Signals and Systems’ class, we are mostly interested in two kinds of signals (or systems): continuous time and discrete time*.

For continuous time signal, its value is well defined over ANY time point you specify as long as it’s within the real line. The time can be , , , , etc. Listing the value at every time point is not an option because there are uncountablely many of them. So we describe continuous signal as a parametric function over , e.g. .

As for discrete time signal (or time series), you can tell the value for sure ONLY at a countable number of time points. Everything in between is up to interpretation, which we will study later when we get to sampling. I didn’t say these time points have to be evenly spaced (uniformly sampled), but they usually are in this class.

We can define a discrete time signal similarly using a parametric function such as , where is the integer time index. But this time, because the number of time points in a discrete time signal is countable, you can choose to build a look-up table (a vector of values, such as ) if your signal is non-zero (non-trivial) over a fixed a finite number of time points (has finite support).

In fact, vectors (aka look-up tables) is the representation your computer uses to store audio recordings: a sound clip has a finite time span and you only need a countable number of time points to play it back (we’ll get to that in sampling theory). You will learn more about it when we get to Digital Signal Processing (DSP).

Under the concept of vector spaces, (parametric) functions and vectors (look-up tables) can be thought of as the same thing. All rules in linear algebra applies, which I will show you later.

Traditionally, discrete time and continuous time systems are clearly separated and taught in any order depending on the teacher’s preference. After you mastered the first one, the rest are almost identical with a few changes.

In my opinion, it is organized like a newbie programmer copying-and-pasting code all over the place when she/he should have refactored it into a reusable function.

‘Ideas Oversimplified’ offers a new approach to learn the essence of the 4-transforms {continuous, discrete}x{Fourier, generalized}, in one shot: using inner products in (linear) vector spaces. More on that later.

* Whenever I say ‘time’ (), it can be replaced by any parameter, such as distance (). It’s just a running variable. I chose not to use other than for intro to complex numbers because I want to reserve the alphabet and for input and the output .